The space of Whitney levels

Abstract

Let X be a continuum and let C( X) be the space of subcontinua of X. In this paper we consider the spaces W( X) = { u: C( X)→ R ∣ u is a Whitney map and u( X) = 1} with the “sup metric” and, N( X)={ u -1( t)∈ C( C( X)): u∈ W( X) and 0⩽ t⩽1}. We define a natural order for N( X) and we prove that if there is a homeomorphism from N( X) onto N( Y) which preserves order, then X is homeomorphic to Y. Also we prove that W( X) is always homeomorphic to l 2 (this answers a question by Nadler).